= Lp norms converge to the supremum norm
{c}
{title2=$\lim_{p\to\infty}\|f\|_p=\|f\|_\infty$}
On a measure space of finite measure, a bounded measurable function satisfies the displayed limit, where the right side is the <essential supremum> of its absolute value. The upper estimate is $\|f\|_p\leq\mu(X)^{1/p}\|f\|_\infty$. For every $a<\|f\|_\infty$, the set where $|f|>a$ has positive measure, giving the reverse limiting estimate. For a continuous function on an open Euclidean domain, the <essential supremum> equals the ordinary supremum. Finiteness of the measure is part of this statement.
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